NEET Physics Nuclei Questions Solved

The electric potential between a proton and an electron is given by $\mathrm{V}={\mathrm{V}}_{0}\mathrm{ln}\frac{\mathrm{r}}{{\mathrm{r}}_{0}}$ where ${\mathrm{r}}_{0}$ is a constant. Assuming Bohr’s model to be applicable, write variation of ${\mathrm{r}}_{\mathrm{n}}$ with n, n being the principal quantum number

(a) ${\mathrm{r}}_{\mathrm{n}}\propto \mathrm{n}$            (b)  ${\mathrm{r}}_{\mathrm{n}}\propto 1/\mathrm{n}$
(c) ${\mathrm{r}}_{\mathrm{n}}\propto {\mathrm{n}}^{2}$           (d) ${\mathrm{r}}_{\mathrm{n}}\propto 1/{\mathrm{n}}^{2}$

(a) Potential energy $\mathrm{U}=\mathrm{eV}={\mathrm{eV}}_{0}\mathrm{ln}\frac{\mathrm{r}}{{\mathrm{r}}_{0}}$
$\therefore$ Force $\mathrm{F}=-\left|\frac{\mathrm{dU}}{\mathrm{dr}}\right|=\frac{{\mathrm{eV}}_{0}}{\mathrm{r}}$ .
$\therefore$ The force will provide the necessary centripetal force. Hence $\frac{{\mathrm{mv}}^{2}}{\mathrm{r}}=\frac{{\mathrm{eV}}_{0}}{\mathrm{r}}⇒\mathrm{v}=\sqrt{\frac{{\mathrm{eV}}_{0}}{\mathrm{m}}}$ …..(i)
and $\mathrm{mvr}=\frac{\mathrm{nh}}{2\mathrm{\pi }}$               …..(ii)
From equation (i) and(ii) $\mathrm{mr}=\left(\frac{\mathrm{nh}}{2\mathrm{\pi }}\right)\sqrt{\frac{\mathrm{m}}{{\mathrm{eV}}_{0}}}$  or r ∝ n

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